SOLUTION: Factoring Completely: How could you factor a problem like this: (a^2 - 10ab + 3b^2)

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Question 159228: Factoring Completely: How could you factor a problem like this: (a^2 - 10ab + 3b^2)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Factoring Completely: How could you factor a problem like this: (a^2 - 10ab + 3b^2)
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The factors will have the form (a - Rb)*(a - Sb) where R and S are to be found.
R*S = +3 and R+S = -10. R and S have to be negative, since their sum is negative and their product is positive.
There are no integers that fit.
The values can be found by using the quadratic eqn:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-10%29%5E2-4%2A1%2A3=88.

Discriminant d=88 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+88+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+88+%29%29%2F2%5C1+=+9.69041575982343
x%5B2%5D+=+%28-%28-10%29-sqrt%28+88+%29%29%2F2%5C1+=+0.30958424017657

Quadratic expression 1x%5E2%2B-10x%2B3 can be factored:
1x%5E2%2B-10x%2B3+=+%28x-9.69041575982343%29%2A%28x-0.30958424017657%29
Again, the answer is: 9.69041575982343, 0.30958424017657. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B3+%29

The values are R = (5 + sqrt(22)) and S = (5 - sqrt(22))
Sub those for R and S into (a - Rb)*(a - Sb)